STA 235H - Interpretation Review

class: center, middle, inverse, title-slide

.title[
# STA 235H - Interpretation Review
]
.subtitle[
## Fall 2023
]
.author[
### McCombs School of Business, UT Austin
]

---

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# Structure

- We will review **.darkorange[different types of regressions]**

- I will provide a general framework of **.darkorange[how to think about these regressions]**

- Try to do the exercises **.darkorange[by yourself]** (*don't cheat!*).

- **.darkorange[Answers]** are posted at the end.

---
# Typical regression: Continuous variables

`$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$`

- `$Y$` is a continuous variable

- `$X$` is a continuous variable

.center[
*A one-unit increase in `$X_1$` is associated with an average increase in `$Y$` of `$\beta_1$` units, holding `$X_2$` constant*
]

---
# Typical regression: Example

Let's analyze what factors associate to someone's grade:

`$$Grade = \beta_0 + \beta_1 HrsAssign + \beta_2PreGrade + \beta_3ClassAtt + \varepsilon$$`

Where:

- **Grade**: Grade in an assignment (out of 100 points).
- **HrsAssign**: Hours dedicated to the assignment.
- **PreGrade**: Grade in the previous assignment (out of 100 points).
- **ClassAtt**: Whether someone attended the last class (1) or not (0).

.box-5trans[Interpret the coefficient for Hours]

---
# Typical regression: Example

.small[

```r
summary(lm(grade ~ hours + pregrade + classatt, data = d))
```

```
## 
## Call:
## lm(formula = grade ~ hours + pregrade + classatt, data = d)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -2.59771 -0.62967 0.07248 0.57736 2.42308 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 24.22912 1.12686 21.50 < 2e-16 ***
## hours 1.89525 0.02079 91.16 < 2e-16 ***
## pregrade 0.44563 0.01227 36.32 < 2e-16 ***
## classatt 1.04993 0.14461 7.26 8.85e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9254 on 196 degrees of freedom
## Multiple R-squared: 0.9808,	Adjusted R-squared: 0.9805 
## F-statistic: 3338 on 3 and 196 DF, p-value: < 2.2e-16
```
]

---
# Typical regression: Binary covariate

`$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$`

- `$Y$` is a continuous variable

- `$X$` is a binary variable

.center[
The event of `$X_1=1$` is associated with an average increase in `$Y$` of `$\beta_1$` units compared to `$X_1=0$`, holding `$X_2$` constant
]

---
# Typical regression: Example

Let's analyze what factors associate to someone's grade:

`$$Grade = \beta_0 + \beta_1 HrsAssign + \beta_2PreGrade + \beta_3ClassAtt + \varepsilon$$`

Where:

.box-5trans[Interpret the coefficient for Class Attendance]

---
# Typical regression: Example

- Interpret the coefficient for *ClassAtt*:

.small[

```r
summary(lm(grade ~ hours + pregrade + classatt, data = d))
```

---
# Regressions with logarithms: Log-level

`$$\log(Y) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$`

- The outcome variable is in a logarithm, `$\log(Y)$`

- `$X$` is a continuous variable, not in a logarithm.

.center[
*A one-unit increase in `$X_1$` is associated with an average increase in `$Y$` of `$\beta_1\times100$` percent, holding `$X_2$` constant*
]

---
# Regressions with logarithms: Example

Let's analyze what characteristics are associated with sales on Etsy:

`$$\log(Sales) = \beta_0 + \beta_1 Price + \beta_2AvgRating + \beta_3NReviews + \varepsilon$$`

Where:

- **log(Sales)**: Weekly sales of a product ($), in a logarithm.
- **Price**: Price of the product ($)
- **AvgRating**: Average rating of the product (in a scale fo 1-5).
- **NReviews**: Number of reviews of a product.

.box-5trans[Interpret the coefficient for Number of Reviews]

---
# Regressions with logarithms: Example

- Interpret the coefficient for *NReviews*:

.small[

```r
summary(lm(log(sales) ~ price + avgrating + nreviews, data = d))
```

```
## 
## Call:
## lm(formula = log(sales) ~ price + avgrating + nreviews, data = d)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -0.54727 -0.03702 0.01991 0.06927 0.19050 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 6.3319302 0.0405368 156.202 < 2e-16 ***
## price 0.0004073 0.0006945 0.586 0.558 
## avgrating 0.0381499 0.0077626 4.915 1.48e-06 ***
## nreviews 0.0190180 0.0003177 59.866 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1119 on 296 degrees of freedom
## Multiple R-squared: 0.9261,	Adjusted R-squared: 0.9253 
## F-statistic: 1236 on 3 and 296 DF, p-value: < 2.2e-16
```
]

---
# Regressions with logarithms: Your turn

- Interpret the coefficient for *Price*:

.small[

```r
summary(lm(log(sales) ~ price + avgrating + nreviews, data = d))
```

---
# Regressions with Interactions

`$$Y = \beta_0 + \beta_1 X_1 + \beta_2D +  \beta_3X_1\times D + \beta_4 X_2 + \varepsilon$$`

- `$Y$` is a continuous variable

- `$X$` is a continuous variable

- `$D$` is a binary variable

.center[
*A one-unit increase in `$X_1$` is associated with an average increase in `$Y$` of `$\beta_1$` units for `$D=0$`, holding `$X_2$` constant*
]

.center[
*A one-unit increase in `$X_1$` is associated with an average increase in `$Y$` of `$\beta_1 + \beta_3$` units for `$D=1$`, holding `$X_2$` constant*
]

---
# Regressions with Interactions: Example

Let's analyze what factors associate to someone's grade:

`$$Grade = \beta_0 + \beta_1PreGrade + \beta_2 HrsAssign + \beta_3ClassAtt + \beta_4HrsAssign \times ClassAtt + \varepsilon$$`

Where:

.box-5trans[Interpret the association between Hours and Grade for students that attended the last class and those that did not]

---
# Regressions with Interactions: Example

- Interpret the association the assicuation between *hours* and *grade* for the two groups.

.small[

```r
summary(lm(grade ~ pregrade + hours*classatt, data = d))
```

```
## 
## Call:
## lm(formula = grade ~ pregrade + hours * classatt, data = d)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1.74995 -0.39230 -0.03793 0.43372 1.99286 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 30.42867 0.88578 34.353 <2e-16 ***
## pregrade 0.35736 0.00957 37.341 <2e-16 ***
## hours 1.65246 0.02769 59.669 <2e-16 ***
## classatt 0.12650 0.22545 0.561 0.575 
## hours:classatt 0.56198 0.03412 16.471 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7216 on 195 degrees of freedom
## Multiple R-squared: 0.9894,	Adjusted R-squared: 0.9892 
## F-statistic: 4540 on 4 and 195 DF, p-value: < 2.2e-16
```
]
---
# Quadratic Regressions

`$$Y = \beta_0 + \beta_1 X_1 + \beta_2X_1^2 + \beta_3 X_2 + \varepsilon$$`

- `$Y$` is a continuous variable

- `$X$` is a continuous variable and includes a quadratic term

.center[
*Increasing `$X_1$` in one unit, from `$x_0$` to `$x_0+1$`, is associated with an average increase in Y of `$2\beta_2 x_0 + \beta_1$` units, holding `$X_2$` constant*
]

---
# Quadratic Regressions: Example

We think there is a quadratic association between price and sales, so we fit the following model:

`$$Sales = \beta_0 + \beta_1 Price + \beta_2Price^2  + \beta_3AvgRating + \beta_4NReviews + \varepsilon$$`

Where:

- **Sales**: Weekly sales of a product ($)
- **Price**: Price of the product ($)
- **AvgRating**: Average rating of the product (in a scale fo 1-5).
- **NReviews**: Number of reviews of a product.

.box-5trans[Interpret the association between Sales and Price, for an increase in Price from &dollar;15 to &dollar;16]

---
# Quadratic Regressions: Example

- Interpret the association between *sales* and *price* (for an increase in price from 15 to 16 dollars)

.small[

```r
summary(lm(sales ~ price + I(price^2) + avgrating + nreviews, data = d))
```

```
## 
## Call:
## lm(formula = sales ~ price + I(price^2) + avgrating + nreviews, 
## data = d)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -238.849 -52.604 5.526 53.160 228.929 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 106.13919 52.09422 2.037 0.0425 * 
## price -10.33381 2.58888 -3.992 8.29e-05 ***
## I(price^2) 2.50065 0.03527 70.901 < 2e-16 ***
## avgrating 50.24900 5.42595 9.261 < 2e-16 ***
## nreviews 30.03823 0.22195 135.337 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 78.17 on 295 degrees of freedom
## Multiple R-squared: 0.9981,	Adjusted R-squared: 0.998 
## F-statistic: 3.781e+04 on 4 and 295 DF, p-value: < 2.2e-16
```
]

---
# Linear Probability Model

`$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$`

- `$Y$` is a binary variable

- `$X$` is a continuous variable

.center[
*Increasing `$X_1$` in one unit is associated with an average increase in the probability of `$Y=1$` of `$\beta_1$`, holding `$X_2$` constant*
]

.center[
*Increasing `$X_1$` in one unit is associated with an average increase in the probability of `$Y=1$` of `$\beta_1\times100$` percentage points, holding `$X_2$` constant*
]

---
# Linear Probability Model: Example

Let's analyze what factors associate to getting an A in an assignment:

`$$Grade = \beta_0 + \beta_1 HrsAssign + \beta_2PreGrade + \beta_3ClassAtt + \varepsilon$$`

Where:

- **GradeA**: Binary variable if the grade in the assignment is A (1) or not (0).
- **HrsAssign**: Hours dedicated to the assignment.
- **PreGrade**: Grade in the previous assignment (out of 100 points).
- **ClassAtt**: Whether someone attended the last class (1) or not (0).

.box-5trans[Interpret the coefficient for Hours]

---
# Linear Probability Model: Example

- Interpret the coefficient for *Hours*.

.small[

```r
summary(lm_robust(gradeA ~ hours + pregrade + classatt, data = d))
```

```
## 
## Call:
## lm_robust(formula = gradeA ~ hours + pregrade + classatt, data = d)
## 
## Standard error type:  HC2 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  CI Lower  CI Upper  DF
## (Intercept) -0.516716   0.208172  -2.482 0.013900 -0.927261 -0.106171 196
## hours        0.020050   0.007420   2.702 0.007488  0.005418  0.034683 196
## pregrade     0.004937   0.001993   2.477 0.014090  0.001006  0.008867 196
## classatt    -0.026360   0.025733  -1.024 0.306914 -0.077109  0.024388 196
## 
## Multiple R-squared:  0.2117 ,	Adjusted R-squared:  0.1997 
## F-statistic: 2.486 on 3 and 196 DF,  p-value: 0.06188
```
]

---
# Answers
.small[
1. A one-hour increase in studying is associated with an average 1.9 point increase in the assignment grade, holding other variables constant.

2. Attending the last class is associated with an average increase in grade of 1.05 points compared to not attending, holding other variables constant.

3. Having one additional review is associated with an average increase in sales of 1.9 percent, holding price and average rating constant.

4. One additional hour dedicated to the assignment is associated to an average increase of 1.65 points in the homework score for students that didn't attend the last class, holding previous grade constant.

5. One additional hour dedicated to the assignment is associated to an average increase of 2.2 points in the homework score for students that attended the last class, holding previous grade constant.

6. Increasing the sales price from 15 to 16 dollars is associated to an average increase in sales of &dollar;64.7, holding other variables constant.

7. One additional hour dedicated to the homework is associated to an average increase in the probability of getting an A of 2 percentage points, holding other variables constant.
]